The council believes that it is the first in the UK to seek such wide expressions of interest from the third sector such as volunteers and friends groups in running "one, five or 10" parks, said head of parks David Allister.
Advertisements ran in local newspapers and around two-dozen local groups expressed interest, as well as one national charity, he said. Richmond was fortunate in having close relationships with hands-on friends groups, he added.
"This is the next logical step but they are expressions of interest only - none of them may go anywhere or all of them may go everywhere," said Allister. "What we won't do is compromise on park quality. It has to be maintained or improved."
He voiced the view that some of the groups could improve quality because they are dedicated to only one park whereas the council has to look after them all.
Elsewhere, some authorities have leased the occasional green space to a closely-related group, said Allister. "I think we are the first to advertise to anyone from small groups to national organisations interested in taking over a whole raft of parks. I work for local government and local government is for local people.
"There is a logic to this initiative if done properly. We won't just hand over the keys to a green space and let an organisation put up a fun fair," he added.